\section{Introduction}

Let $G = (V, E)$ be an undirected graph. A \emph{stable set} in $G$ is a subset of pairwise non-adjacent vertices of $G$. 
%Given a graph $G$, the \emph{maximum cardinality stable set problem} (MSS) asks
% for a stable set $S$ in $G$ of maximum cardinality. 
The {\em stability number of $G$} is the cardinality of $S$ and is denoted by $\alpha(G)$. 
%MSS is NP-hard, and has been approached in the literature through several
% techniques. 
Let $n := |V|$ and $\mathcal S(G)\subseteq\{0,1\}^n$ be the set of all
characteristic vectors of stable sets of $G$. The polytope of stable sets of $G$
is denoted by $STAB(G) = \mbox{conv} \{ x \mid x \in \mathcal S(G) \}$.

Several procedures for generating cuts for $STAB(G)$ exist, see, e.g.,
\cite{Pardalos}, \cite{Rossi.Smriglio.01}. In a previous work \cite{Correa14} we
proposed the use of clique projections as a general method for cutting plane
generation for $STAB(G)$, along with a new clique lifting procedure leading to
stronger inequalities than those obtained with previous methods. In this work we
present a strengthening of this method, which allows us to generate additional
facet-inducing inequalities for $STAB(G)$. We also present
sufficient conditions for the valid inequalities generated with
this new method to be facet defining. These conditions slightly
generalize the ones in~\cite{XavierCampelo11}. Computational experience on
random and DIMACS benchmark instances shows that the proposed approach allows to
obtain tighter upper bounds within a reasonable computation time.

